Maximum Likelihood Estimation (MLE) for a Uniform Distribution

Maximum likelihood estimation for uniform distribution

A uniform distribution is a probability distribution in which every value between an interval from to is equally likely to be chosen.

The probability that we will obtain a value between x1 and x2 on an interval from to can be found using the formula:

P(obtain value between x1 and x2)  =  (x2 – x1) / (b – a)

Uniform distribution example

This tutorial explains how to find the maximum likelihood estimate (mle) for parameters and of the uniform distribution.

Maximum Likelihood Estimation

Step 1: Write the likelihood function.

For a uniform distribution, the likelihood function can be written as:

likelihood function for uniform distribution

Step 2: Write the log-likelihood function. 

Log-likelihood function of uniform distribution

Step 3: Find the values for and that maximize the log-likelihood by taking the derivative of the log-likelihood function with respect to and b.

The derivative of the log-likelihood function with respect to can be written as:

Derivative of log-likelihood function for uniform distribution

Similarly, the derivative of the log-likelihood function with respect to can be written as:

Partial derivative of log-likelihood function for uniform distribution

Step 4: Identify the maximum likelihood estimators for and b.

Notice that the derivative with respect to is monotonically increasing. Thus, the mle for a would be the largest possible, which would simply be:

min(X1, X2, … , Xn)

Also notice that the derivative with respect to is monotonically decreasing. Thus, the mle for would be the smallest possible, which would be:

max(X1, X2, … , Xn)

Additional Resources

An Introduction to the Uniform Distribution
Uniform Distribution Calculator
An Excellent Explanation of Maximum likelihood estimation
Wikipedia: What is a monotonically increasing or decreasing function?

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